= Spherical square-well scattering length
{title2=$a_s=a-\tan(\kappa a)/\kappa$}
For an attractive <spherical square well>, let $\kappa=\sqrt{2mV_0}/\hbar$. At zero <energy>, the outer reduced radial wave is proportional to $r-a_s$ and the regular inner wave to $\sin(\kappa r)$. Matching their <logarithmic derivatives> gives $a_s=a-\tan(\kappa a)/\kappa$. Away from a threshold and at sufficiently low <energy>, the total <S-wave scattering> <scattering cross-section> tends to $4\pi a_s^2$. The length can be negative, zero or divergent.
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